BaselineHazardFunktion

The baseline hazard function, denoted h0(t), is a key concept in survival analysis, particularly in the Cox proportional hazards model. It represents the instantaneous risk of the event at time t when all covariates are at their reference (often zero) levels. The model expresses the individual hazard as h(t|X) = h0(t) exp(β^T X), where X is a vector of covariates and β are the log-relative risk coefficients. Thus the baseline hazard captures how the hazard evolves over time independently of covariate effects.

The baseline cumulative hazard function, H0(t) = ∫0^t h0(u) du, summarizes the total hazard accumulated up to time t. In the Cox model, β is typically estimated without specifying h0(t) (via partial likelihood), leaving h0(t) unspecified. The baseline hazard or its cumulative form can then be estimated nonparametrically from the data. The most common nonparametric estimator of the baseline cumulative hazard is the Breslow estimator, which updates at each observed event time t_j as ΔH0(t_j) = d_j / Σ_{i in R_j} exp(β^T X_i); here d_j is the number of events at t_j and R_j is the risk set just before t_j. The baseline hazard h0(t) is then interpreted as the size of these jumps divided by the corresponding risk set, forming a step function.

Applications include deriving the baseline survival function, S0(t) = exp(-H0(t)), and using it to obtain an individual survival function S(t|X) = exp(-H0(t) exp(β^T X)). Parametric survival models, by contrast, impose a specified form on h0(t) (e.g., Weibull or Gompertz). The baseline hazard thus provides a time-specific reference against which covariate effects are measured.